Poisson algebras in reduction of symmetries

We investigate four Poisson algebras occurring in the reduction of symmetries of Hamiltonian systems. We show that all four algebras are naturally isomorphic if the action of the symmetry group is free and proper. For a compact group action reduced at the zero level of the momentum map, two pairs of the algebras are naturally isomorphic. We give examples in which all four algebras are not isomorphic.